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Shared Qs (031)


  1. Question

    Integration reverses differentiation. We will practice “undoing” the power rule by finding an antiderivative.

    Let \(f(x)\) be an unknown polynomial. Its derivative is known.

    \[f'(x)={-15}x^{4}+{16}x^{3}-{18}x^{2}\]

    To undo the power rule, each term’s power is increased by 1, and then the coefficient is divided by the new power. This gets us most of the way.

    A (slightly annoying) complication occurs because adding any constant to \(f(x)\) would not change its derivative, since the derivative of a constant is 0. This means there are infinitely many possible antiderivatives with a constant of integration equaling any real number. If we know a single solution to \(y=f(x)\) (a single point on the curve), then the constant of integration can be determined. In this case, let’s enforce the following condition:

    \[f(2)=-84\]

    We know the integral can be expressed as a polynomial with 4 terms, where \(j>k>l\):

    \[f(x)=Jx^{j}+Kx^{k}+Lx^l+C\]

    Find the parameters.



    Solution


  2. Question

    Integration reverses differentiation. We will practice “undoing” the power rule by finding an antiderivative.

    Let \(f(x)\) be an unknown polynomial. Its derivative is known.

    \[f'(x)={-10}x^{4}-{32}x^{3}-{4}{}\]

    To undo the power rule, each term’s power is increased by 1, and then the coefficient is divided by the new power. This gets us most of the way.

    A (slightly annoying) complication occurs because adding any constant to \(f(x)\) would not change its derivative, since the derivative of a constant is 0. This means there are infinitely many possible antiderivatives with a constant of integration equaling any real number. If we know a single solution to \(y=f(x)\) (a single point on the curve), then the constant of integration can be determined. In this case, let’s enforce the following condition:

    \[f(-4)=11\]

    We know the integral can be expressed as a polynomial with 4 terms, where \(j>k>l\):

    \[f(x)=Jx^{j}+Kx^{k}+Lx^l+C\]

    Find the parameters.



    Solution


  3. Question

    Integration reverses differentiation. We will practice “undoing” the power rule by finding an antiderivative.

    Let \(f(x)\) be an unknown polynomial. Its derivative is known.

    \[f'(x)={-12}x^{3}+{24}x^{2}-{18}x^{}\]

    To undo the power rule, each term’s power is increased by 1, and then the coefficient is divided by the new power. This gets us most of the way.

    A (slightly annoying) complication occurs because adding any constant to \(f(x)\) would not change its derivative, since the derivative of a constant is 0. This means there are infinitely many possible antiderivatives with a constant of integration equaling any real number. If we know a single solution to \(y=f(x)\) (a single point on the curve), then the constant of integration can be determined. In this case, let’s enforce the following condition:

    \[f(2)=-26\]

    We know the integral can be expressed as a polynomial with 4 terms, where \(j>k>l\):

    \[f(x)=Jx^{j}+Kx^{k}+Lx^l+C\]

    Find the parameters.



    Solution


  4. Question

    Integration reverses differentiation. We will practice “undoing” the power rule by finding an antiderivative.

    Let \(f(x)\) be an unknown polynomial. Its derivative is known.

    \[f'(x)={-35}x^{6}-{36}x^{5}+{35}x^{4}\]

    To undo the power rule, each term’s power is increased by 1, and then the coefficient is divided by the new power. This gets us most of the way.

    A (slightly annoying) complication occurs because adding any constant to \(f(x)\) would not change its derivative, since the derivative of a constant is 0. This means there are infinitely many possible antiderivatives with a constant of integration equaling any real number. If we know a single solution to \(y=f(x)\) (a single point on the curve), then the constant of integration can be determined. In this case, let’s enforce the following condition:

    \[f(-2)=39\]

    We know the integral can be expressed as a polynomial with 4 terms, where \(j>k>l\):

    \[f(x)=Jx^{j}+Kx^{k}+Lx^l+C\]

    Find the parameters.



    Solution


  5. Question

    Integration reverses differentiation. We will practice “undoing” the power rule by finding an antiderivative.

    Let \(f(x)\) be an unknown polynomial. Its derivative is known.

    \[f'(x)={20}x^{4}-{12}x^{3}-{27}x^{2}\]

    To undo the power rule, each term’s power is increased by 1, and then the coefficient is divided by the new power. This gets us most of the way.

    A (slightly annoying) complication occurs because adding any constant to \(f(x)\) would not change its derivative, since the derivative of a constant is 0. This means there are infinitely many possible antiderivatives with a constant of integration equaling any real number. If we know a single solution to \(y=f(x)\) (a single point on the curve), then the constant of integration can be determined. In this case, let’s enforce the following condition:

    \[f(2)=2\]

    We know the integral can be expressed as a polynomial with 4 terms, where \(j>k>l\):

    \[f(x)=Jx^{j}+Kx^{k}+Lx^l+C\]

    Find the parameters.



    Solution


  6. Question

    A definite integral can be used to express the area under a curve. For example, say we wanted to find the area under \(f(x)=x^{}-x^{7}\) between \(x=0.43\) and \(x=0.81\). Let’s draw a picture.

    plot of chunk unnamed-chunk-2

    The area equals the definite integral of the function on interval [0.43, 0.81].

    \[A ~=~ \int_{0.43}^{0.81} \left(x^{}-x^{7} \right)~dx\]

    This definite integral can be evaluated from an antiderivative of \(f(x)\). Let \(g(x)\) be an antiderivative of \(f(x)\) such that \(g'(x)=f(x)\).

    \[A~=~g(0.81)-g(0.43)\]

    Find the area. The tolerance is \(\pm 0.001\).


    Solution


  7. Question

    A definite integral can be used to express the area under a curve. For example, say we wanted to find the area under \(f(x)=x^{3}-x^{9}\) between \(x=0.51\) and \(x=0.8\). Let’s draw a picture.

    plot of chunk unnamed-chunk-2

    The area equals the definite integral of the function on interval [0.51, 0.8].

    \[A ~=~ \int_{0.51}^{0.8} \left(x^{3}-x^{9} \right)~dx\]

    This definite integral can be evaluated from an antiderivative of \(f(x)\). Let \(g(x)\) be an antiderivative of \(f(x)\) such that \(g'(x)=f(x)\).

    \[A~=~g(0.8)-g(0.51)\]

    Find the area. The tolerance is \(\pm 0.001\).


    Solution


  8. Question

    A definite integral can be used to express the area under a curve. For example, say we wanted to find the area under \(f(x)=x^{}-x^{6}\) between \(x=0.49\) and \(x=0.87\). Let’s draw a picture.

    plot of chunk unnamed-chunk-2

    The area equals the definite integral of the function on interval [0.49, 0.87].

    \[A ~=~ \int_{0.49}^{0.87} \left(x^{}-x^{6} \right)~dx\]

    This definite integral can be evaluated from an antiderivative of \(f(x)\). Let \(g(x)\) be an antiderivative of \(f(x)\) such that \(g'(x)=f(x)\).

    \[A~=~g(0.87)-g(0.49)\]

    Find the area. The tolerance is \(\pm 0.001\).


    Solution


  9. Question

    A definite integral can be used to express the area under a curve. For example, say we wanted to find the area under \(f(x)=x^{}-x^{5}\) between \(x=0.54\) and \(x=0.8\). Let’s draw a picture.

    plot of chunk unnamed-chunk-2

    The area equals the definite integral of the function on interval [0.54, 0.8].

    \[A ~=~ \int_{0.54}^{0.8} \left(x^{}-x^{5} \right)~dx\]

    This definite integral can be evaluated from an antiderivative of \(f(x)\). Let \(g(x)\) be an antiderivative of \(f(x)\) such that \(g'(x)=f(x)\).

    \[A~=~g(0.8)-g(0.54)\]

    Find the area. The tolerance is \(\pm 0.001\).


    Solution


  10. Question

    A definite integral can be used to express the area under a curve. For example, say we wanted to find the area under \(f(x)=x^{3}-x^{7}\) between \(x=0.56\) and \(x=0.85\). Let’s draw a picture.

    plot of chunk unnamed-chunk-2

    The area equals the definite integral of the function on interval [0.56, 0.85].

    \[A ~=~ \int_{0.56}^{0.85} \left(x^{3}-x^{7} \right)~dx\]

    This definite integral can be evaluated from an antiderivative of \(f(x)\). Let \(g(x)\) be an antiderivative of \(f(x)\) such that \(g'(x)=f(x)\).

    \[A~=~g(0.85)-g(0.56)\]

    Find the area. The tolerance is \(\pm 0.001\).


    Solution


  11. Question

    A particle moves up and down along a 1-dimensional path. It’s acceleration (\(a(t)\) in \(\mathrm{\frac{m}{s^2}}\)) is a function of time (\(t\) in seconds).

    \[a(t)=-48t+12\]

    The particle’s velocity is also a function of time. Also, \(v(0)=4\).

    \[v(t) ~=~ 4 + \int_{0}^{t}a(t)\,dt\]

    The particle’s position is also a function of time. Also, \(x(0)=4\).

    \[x(t) ~=~ 4 + \int_{0}^{t}v(t)\,dt\]

    Evaluate position, velocity, and acceleration at \(t=0\), \(t=1\), and \(t=2\).

    t y(t) v(t) a(t)
    0
    1
    2


    Solution


  12. Question

    A particle moves up and down along a 1-dimensional path. It’s acceleration (\(a(t)\) in \(\mathrm{\frac{m}{s^2}}\)) is a function of time (\(t\) in seconds).

    \[a(t)=-36t+10\]

    The particle’s velocity is also a function of time. Also, \(v(0)=8\).

    \[v(t) ~=~ 8 + \int_{0}^{t}a(t)\,dt\]

    The particle’s position is also a function of time. Also, \(x(0)=2\).

    \[x(t) ~=~ 2 + \int_{0}^{t}v(t)\,dt\]

    Evaluate position, velocity, and acceleration at \(t=0\), \(t=1\), and \(t=2\).

    t y(t) v(t) a(t)
    0
    1
    2


    Solution


  13. Question

    A particle moves up and down along a 1-dimensional path. It’s acceleration (\(a(t)\) in \(\mathrm{\frac{m}{s^2}}\)) is a function of time (\(t\) in seconds).

    \[a(t)=-48t+20\]

    The particle’s velocity is also a function of time. Also, \(v(0)=6\).

    \[v(t) ~=~ 6 + \int_{0}^{t}a(t)\,dt\]

    The particle’s position is also a function of time. Also, \(x(0)=-1\).

    \[x(t) ~=~ -1 + \int_{0}^{t}v(t)\,dt\]

    Evaluate position, velocity, and acceleration at \(t=0\), \(t=1\), and \(t=2\).

    t y(t) v(t) a(t)
    0
    1
    2


    Solution


  14. Question

    A particle moves up and down along a 1-dimensional path. It’s acceleration (\(a(t)\) in \(\mathrm{\frac{m}{s^2}}\)) is a function of time (\(t\) in seconds).

    \[a(t)=-36t+4\]

    The particle’s velocity is also a function of time. Also, \(v(0)=8\).

    \[v(t) ~=~ 8 + \int_{0}^{t}a(t)\,dt\]

    The particle’s position is also a function of time. Also, \(x(0)=10\).

    \[x(t) ~=~ 10 + \int_{0}^{t}v(t)\,dt\]

    Evaluate position, velocity, and acceleration at \(t=0\), \(t=1\), and \(t=2\).

    t y(t) v(t) a(t)
    0
    1
    2


    Solution


  15. Question

    A particle moves up and down along a 1-dimensional path. It’s acceleration (\(a(t)\) in \(\mathrm{\frac{m}{s^2}}\)) is a function of time (\(t\) in seconds).

    \[a(t)=-54t+14\]

    The particle’s velocity is also a function of time. Also, \(v(0)=9\).

    \[v(t) ~=~ 9 + \int_{0}^{t}a(t)\,dt\]

    The particle’s position is also a function of time. Also, \(x(0)=2\).

    \[x(t) ~=~ 2 + \int_{0}^{t}v(t)\,dt\]

    Evaluate position, velocity, and acceleration at \(t=0\), \(t=1\), and \(t=2\).

    t y(t) v(t) a(t)
    0
    1
    2


    Solution


  16. Question

    A 3D shape’s base is the 2D region enclosed by \(y=0\) and \(y=f(x)=x^{0.8}-x^{5.5}\) on the \(xy\) plane. For every plane perpendicular to the \(x\) axis, with \(0<x<1\), there is a square cross section.

    I have attempted to draw this below.

    plot of chunk unnamed-chunk-2

    And here is a spinning animation. Does that help?

    plot of chunk unnamed-chunk-3

    Find the volume of the shape; the tolerance is \(\pm 0.001\) cubic units. You might find this video helpful. The general topic is “volume from cross sections”; you might try a variety of resources by searching this phrase in your favorite search engine.


    Solution


  17. Question

    A 3D shape’s base is the 2D region enclosed by \(y=0\) and \(y=f(x)=x^{0.5}-x^{2.4}\) on the \(xy\) plane. For every plane perpendicular to the \(x\) axis, with \(0<x<1\), there is a square cross section.

    I have attempted to draw this below.

    plot of chunk unnamed-chunk-2

    And here is a spinning animation. Does that help?

    plot of chunk unnamed-chunk-3

    Find the volume of the shape; the tolerance is \(\pm 0.001\) cubic units. You might find this video helpful. The general topic is “volume from cross sections”; you might try a variety of resources by searching this phrase in your favorite search engine.


    Solution


  18. Question

    A 3D shape’s base is the 2D region enclosed by \(y=0\) and \(y=f(x)=x^{0.7}-x^{8.6}\) on the \(xy\) plane. For every plane perpendicular to the \(x\) axis, with \(0<x<1\), there is a square cross section.

    I have attempted to draw this below.

    plot of chunk unnamed-chunk-2

    And here is a spinning animation. Does that help?

    plot of chunk unnamed-chunk-3

    Find the volume of the shape; the tolerance is \(\pm 0.001\) cubic units. You might find this video helpful. The general topic is “volume from cross sections”; you might try a variety of resources by searching this phrase in your favorite search engine.


    Solution


  19. Question

    A 3D shape’s base is the 2D region enclosed by \(y=0\) and \(y=f(x)=x^{1.7}-x^{9.3}\) on the \(xy\) plane. For every plane perpendicular to the \(x\) axis, with \(0<x<1\), there is a square cross section.

    I have attempted to draw this below.

    plot of chunk unnamed-chunk-2

    And here is a spinning animation. Does that help?

    plot of chunk unnamed-chunk-3

    Find the volume of the shape; the tolerance is \(\pm 0.001\) cubic units. You might find this video helpful. The general topic is “volume from cross sections”; you might try a variety of resources by searching this phrase in your favorite search engine.


    Solution


  20. Question

    A 3D shape’s base is the 2D region enclosed by \(y=0\) and \(y=f(x)=x^{1.1}-x^{3.6}\) on the \(xy\) plane. For every plane perpendicular to the \(x\) axis, with \(0<x<1\), there is a square cross section.

    I have attempted to draw this below.

    plot of chunk unnamed-chunk-2

    And here is a spinning animation. Does that help?

    plot of chunk unnamed-chunk-3

    Find the volume of the shape; the tolerance is \(\pm 0.001\) cubic units. You might find this video helpful. The general topic is “volume from cross sections”; you might try a variety of resources by searching this phrase in your favorite search engine.


    Solution


  21. Question

    A 3D shape is produced with elliptical cross sections. As \(x\) progresses from along the interval \([0,1]\), the cross section perpendicular to the \(x\) axis will be an ellipse with axes (and radii) parallel to the \(y\) and \(z\) axes. The (maximum and minimum) radii are power functions with respect to \(x\).

    \[r_1(x) ~=~ x^{1.45}\] \[r_2(x) ~=~ x^{0.38}\]

    plot of chunk unnamed-chunk-2

    plot of chunk unnamed-chunk-3

    Find the volume of the shape. The tolerance is \(\pm 0.01\) cubic units.

    As a hint, I’ll remind you that the area of an ellipse is found by multiplying the radii by each other and pi: \(A=\pi\cdot r_1 \cdot r_2\).


    Solution


  22. Question

    A 3D shape is produced with elliptical cross sections. As \(x\) progresses from along the interval \([0,1]\), the cross section perpendicular to the \(x\) axis will be an ellipse with axes (and radii) parallel to the \(y\) and \(z\) axes. The (maximum and minimum) radii are power functions with respect to \(x\).

    \[r_1(x) ~=~ x^{1.03}\] \[r_2(x) ~=~ x^{2.52}\]

    plot of chunk unnamed-chunk-2

    plot of chunk unnamed-chunk-3

    Find the volume of the shape. The tolerance is \(\pm 0.01\) cubic units.

    As a hint, I’ll remind you that the area of an ellipse is found by multiplying the radii by each other and pi: \(A=\pi\cdot r_1 \cdot r_2\).


    Solution


  23. Question

    A 3D shape is produced with elliptical cross sections. As \(x\) progresses from along the interval \([0,1]\), the cross section perpendicular to the \(x\) axis will be an ellipse with axes (and radii) parallel to the \(y\) and \(z\) axes. The (maximum and minimum) radii are power functions with respect to \(x\).

    \[r_1(x) ~=~ x^{0.67}\] \[r_2(x) ~=~ x^{1}\]

    plot of chunk unnamed-chunk-2

    plot of chunk unnamed-chunk-3

    Find the volume of the shape. The tolerance is \(\pm 0.01\) cubic units.

    As a hint, I’ll remind you that the area of an ellipse is found by multiplying the radii by each other and pi: \(A=\pi\cdot r_1 \cdot r_2\).


    Solution


  24. Question

    A 3D shape is produced with elliptical cross sections. As \(x\) progresses from along the interval \([0,1]\), the cross section perpendicular to the \(x\) axis will be an ellipse with axes (and radii) parallel to the \(y\) and \(z\) axes. The (maximum and minimum) radii are power functions with respect to \(x\).

    \[r_1(x) ~=~ x^{2.11}\] \[r_2(x) ~=~ x^{1.58}\]

    plot of chunk unnamed-chunk-2

    plot of chunk unnamed-chunk-3

    Find the volume of the shape. The tolerance is \(\pm 0.01\) cubic units.

    As a hint, I’ll remind you that the area of an ellipse is found by multiplying the radii by each other and pi: \(A=\pi\cdot r_1 \cdot r_2\).


    Solution


  25. Question

    A 3D shape is produced with elliptical cross sections. As \(x\) progresses from along the interval \([0,1]\), the cross section perpendicular to the \(x\) axis will be an ellipse with axes (and radii) parallel to the \(y\) and \(z\) axes. The (maximum and minimum) radii are power functions with respect to \(x\).

    \[r_1(x) ~=~ x^{1.86}\] \[r_2(x) ~=~ x^{1.64}\]

    plot of chunk unnamed-chunk-2

    plot of chunk unnamed-chunk-3

    Find the volume of the shape. The tolerance is \(\pm 0.01\) cubic units.

    As a hint, I’ll remind you that the area of an ellipse is found by multiplying the radii by each other and pi: \(A=\pi\cdot r_1 \cdot r_2\).


    Solution


  26. Question

    A bowl is designed as the revolution of the region between 3 curves: \[z=4x-44\]

    \[z=\frac{4}{144}x^2\]

    \[z=0\]

    The revolution occurs around the \(z\) axis. The bowl will be made of wood.

    plot of chunk unnamed-chunk-2

    Notice the point of intersection is \((12, 4)\). Below is a wireframe animation of the bowl.

    plot of chunk unnamed-chunk-3

    Find the volume of the wood composing the bowl. I’d recommend using the washer method. The tolerance is \(\pm 1\) cubic units.


    Solution


  27. Question

    A bowl is designed as the revolution of the region between 3 curves: \[z=4x-30\]

    \[z=\frac{6}{81}x^2\]

    \[z=0\]

    The revolution occurs around the \(z\) axis. The bowl will be made of wood.

    plot of chunk unnamed-chunk-2

    Notice the point of intersection is \((9, 6)\). Below is a wireframe animation of the bowl.

    plot of chunk unnamed-chunk-3

    Find the volume of the wood composing the bowl. I’d recommend using the washer method. The tolerance is \(\pm 1\) cubic units.


    Solution


  28. Question

    A bowl is designed as the revolution of the region between 3 curves: \[z=3x-25\]

    \[z=\frac{8}{121}x^2\]

    \[z=0\]

    The revolution occurs around the \(z\) axis. The bowl will be made of wood.

    plot of chunk unnamed-chunk-2

    Notice the point of intersection is \((11, 8)\). Below is a wireframe animation of the bowl.

    plot of chunk unnamed-chunk-3

    Find the volume of the wood composing the bowl. I’d recommend using the washer method. The tolerance is \(\pm 1\) cubic units.


    Solution


  29. Question

    A bowl is designed as the revolution of the region between 3 curves: \[z=2x-11\]

    \[z=\frac{9}{100}x^2\]

    \[z=0\]

    The revolution occurs around the \(z\) axis. The bowl will be made of wood.

    plot of chunk unnamed-chunk-2

    Notice the point of intersection is \((10, 9)\). Below is a wireframe animation of the bowl.

    plot of chunk unnamed-chunk-3

    Find the volume of the wood composing the bowl. I’d recommend using the washer method. The tolerance is \(\pm 1\) cubic units.


    Solution


  30. Question

    A bowl is designed as the revolution of the region between 3 curves: \[z=5x-61\]

    \[z=\frac{4}{169}x^2\]

    \[z=0\]

    The revolution occurs around the \(z\) axis. The bowl will be made of wood.

    plot of chunk unnamed-chunk-2

    Notice the point of intersection is \((13, 4)\). Below is a wireframe animation of the bowl.

    plot of chunk unnamed-chunk-3

    Find the volume of the wood composing the bowl. I’d recommend using the washer method. The tolerance is \(\pm 1\) cubic units.


    Solution


  31. Question

    An isosceles trapezoid is the region between the following curves:

    \[z ~=~ 0.65x\] \[z ~=~ -0.65x\] \[x ~=~ 2.16\] \[x ~=~ 9.38\]

    plot of chunk unnamed-chunk-2

    That trapezoid is revolved around the \(z\) axis to produce a toroid. A wireframe animation of the toroid is shown below.

    plot of chunk unnamed-chunk-3

    Find the volume of the toroid using shell integration. The tolerance is \(\pm 1\) cubic unit.


    Solution


  32. Question

    An isosceles trapezoid is the region between the following curves:

    \[z ~=~ 0.79x\] \[z ~=~ -0.79x\] \[x ~=~ 2.56\] \[x ~=~ 7.07\]

    plot of chunk unnamed-chunk-2

    That trapezoid is revolved around the \(z\) axis to produce a toroid. A wireframe animation of the toroid is shown below.

    plot of chunk unnamed-chunk-3

    Find the volume of the toroid using shell integration. The tolerance is \(\pm 1\) cubic unit.


    Solution


  33. Question

    An isosceles trapezoid is the region between the following curves:

    \[z ~=~ 0.95x\] \[z ~=~ -0.95x\] \[x ~=~ 2.2\] \[x ~=~ 8.74\]

    plot of chunk unnamed-chunk-2

    That trapezoid is revolved around the \(z\) axis to produce a toroid. A wireframe animation of the toroid is shown below.

    plot of chunk unnamed-chunk-3

    Find the volume of the toroid using shell integration. The tolerance is \(\pm 1\) cubic unit.


    Solution


  34. Question

    An isosceles trapezoid is the region between the following curves:

    \[z ~=~ 0.58x\] \[z ~=~ -0.58x\] \[x ~=~ 6.01\] \[x ~=~ 9.16\]

    plot of chunk unnamed-chunk-2

    That trapezoid is revolved around the \(z\) axis to produce a toroid. A wireframe animation of the toroid is shown below.

    plot of chunk unnamed-chunk-3

    Find the volume of the toroid using shell integration. The tolerance is \(\pm 1\) cubic unit.


    Solution


  35. Question

    An isosceles trapezoid is the region between the following curves:

    \[z ~=~ 0.41x\] \[z ~=~ -0.41x\] \[x ~=~ 2.48\] \[x ~=~ 6.08\]

    plot of chunk unnamed-chunk-2

    That trapezoid is revolved around the \(z\) axis to produce a toroid. A wireframe animation of the toroid is shown below.

    plot of chunk unnamed-chunk-3

    Find the volume of the toroid using shell integration. The tolerance is \(\pm 1\) cubic unit.


    Solution


  36. Question

    A bucket filled with water will be pulled up 34 meters from the bottom of a well. The bucket is pulled up by a heavy rope that connects the pulley to the bucket. As the bucket is lifted, this rope’s length decreases from 34 meters to 0 meters. The bucket and water together have a weight of 40 Newtons. The rope has a linear weight density of 5.8 Newtons per meter.

    plot of chunk unnamed-chunk-2

    Each infinitesimal bit of work can be calculated from the product of force times infinitesimal displacement (change in height).

    \[dW = F(x) \, dx\]

    Notice that the force needed to lift the water, without acceleration, equals the total weight of the bucket, water, and rope between the pulley and the bucket. Because the length of rope decreases, the total weight decreases, so it gets easier to lift the bucket each millimeter.

    The total work can be found by adding up all the infinitesimal bits of work. Find the total work in Joules (1 Joule equals 1 Newton\(\cdot\)meter). The tolerance is \(\pm\) 100 J.


    Solution


  37. Question

    A bucket filled with water will be pulled up 14 meters from the bottom of a well. The bucket is pulled up by a heavy rope that connects the pulley to the bucket. As the bucket is lifted, this rope’s length decreases from 14 meters to 0 meters. The bucket and water together have a weight of 41 Newtons. The rope has a linear weight density of 6.8 Newtons per meter.

    plot of chunk unnamed-chunk-2

    Each infinitesimal bit of work can be calculated from the product of force times infinitesimal displacement (change in height).

    \[dW = F(x) \, dx\]

    Notice that the force needed to lift the water, without acceleration, equals the total weight of the bucket, water, and rope between the pulley and the bucket. Because the length of rope decreases, the total weight decreases, so it gets easier to lift the bucket each millimeter.

    The total work can be found by adding up all the infinitesimal bits of work. Find the total work in Joules (1 Joule equals 1 Newton\(\cdot\)meter). The tolerance is \(\pm\) 100 J.


    Solution


  38. Question

    A bucket filled with water will be pulled up 54 meters from the bottom of a well. The bucket is pulled up by a heavy rope that connects the pulley to the bucket. As the bucket is lifted, this rope’s length decreases from 54 meters to 0 meters. The bucket and water together have a weight of 77 Newtons. The rope has a linear weight density of 2.8 Newtons per meter.

    plot of chunk unnamed-chunk-2

    Each infinitesimal bit of work can be calculated from the product of force times infinitesimal displacement (change in height).

    \[dW = F(x) \, dx\]

    Notice that the force needed to lift the water, without acceleration, equals the total weight of the bucket, water, and rope between the pulley and the bucket. Because the length of rope decreases, the total weight decreases, so it gets easier to lift the bucket each millimeter.

    The total work can be found by adding up all the infinitesimal bits of work. Find the total work in Joules (1 Joule equals 1 Newton\(\cdot\)meter). The tolerance is \(\pm\) 100 J.


    Solution


  39. Question

    A bucket filled with water will be pulled up 42 meters from the bottom of a well. The bucket is pulled up by a heavy rope that connects the pulley to the bucket. As the bucket is lifted, this rope’s length decreases from 42 meters to 0 meters. The bucket and water together have a weight of 60 Newtons. The rope has a linear weight density of 5.1 Newtons per meter.

    plot of chunk unnamed-chunk-2

    Each infinitesimal bit of work can be calculated from the product of force times infinitesimal displacement (change in height).

    \[dW = F(x) \, dx\]

    Notice that the force needed to lift the water, without acceleration, equals the total weight of the bucket, water, and rope between the pulley and the bucket. Because the length of rope decreases, the total weight decreases, so it gets easier to lift the bucket each millimeter.

    The total work can be found by adding up all the infinitesimal bits of work. Find the total work in Joules (1 Joule equals 1 Newton\(\cdot\)meter). The tolerance is \(\pm\) 100 J.


    Solution


  40. Question

    A bucket filled with water will be pulled up 13 meters from the bottom of a well. The bucket is pulled up by a heavy rope that connects the pulley to the bucket. As the bucket is lifted, this rope’s length decreases from 13 meters to 0 meters. The bucket and water together have a weight of 60 Newtons. The rope has a linear weight density of 8.5 Newtons per meter.

    plot of chunk unnamed-chunk-2

    Each infinitesimal bit of work can be calculated from the product of force times infinitesimal displacement (change in height).

    \[dW = F(x) \, dx\]

    Notice that the force needed to lift the water, without acceleration, equals the total weight of the bucket, water, and rope between the pulley and the bucket. Because the length of rope decreases, the total weight decreases, so it gets easier to lift the bucket each millimeter.

    The total work can be found by adding up all the infinitesimal bits of work. Find the total work in Joules (1 Joule equals 1 Newton\(\cdot\)meter). The tolerance is \(\pm\) 100 J.


    Solution


  41. Question

    A function \(f(x)\) is graphed below.

    plot of chunk unnamed-chunk-2

    Evaluate the following integral. \[\int_{1}^{10}f(x)\,dx\]


    Solution


  42. Question

    A function \(f(x)\) is graphed below.

    plot of chunk unnamed-chunk-2

    Evaluate the following integral. \[\int_{1}^{7}f(x)\,dx\]


    Solution


  43. Question

    A function \(f(x)\) is graphed below.

    plot of chunk unnamed-chunk-2

    Evaluate the following integral. \[\int_{1}^{5}f(x)\,dx\]


    Solution


  44. Question

    A function \(f(x)\) is graphed below.

    plot of chunk unnamed-chunk-2

    Evaluate the following integral. \[\int_{0}^{10}f(x)\,dx\]


    Solution


  45. Question

    A function \(f(x)\) is graphed below.

    plot of chunk unnamed-chunk-2

    Evaluate the following integral. \[\int_{3}^{8}f(x)\,dx\]


    Solution


  46. Question

    A function \(f(x)\) is graphed below.

    plot of chunk unnamed-chunk-2

    Evaluate the following integral. \[\int_{2}^{8}f(x)\,dx\]


    Solution


  47. Question

    A function \(f(x)\) is graphed below.

    plot of chunk unnamed-chunk-2

    Evaluate the following integral. \[\int_{2}^{9}f(x)\,dx\]


    Solution


  48. Question

    A function \(f(x)\) is graphed below.

    plot of chunk unnamed-chunk-2

    Evaluate the following integral. \[\int_{1}^{7}f(x)\,dx\]


    Solution


  49. Question

    A function \(f(x)\) is graphed below.

    plot of chunk unnamed-chunk-2

    Evaluate the following integral. \[\int_{1}^{6}f(x)\,dx\]


    Solution


  50. Question

    A function \(f(x)\) is graphed below.

    plot of chunk unnamed-chunk-2

    Evaluate the following integral. \[\int_{1}^{9}f(x)\,dx\]


    Solution